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>maybe a sufficiently powerful computer running a sufficiently sophisticated program can be conscious.

Maybe it can. People presenting the hard problem of consciousness do not usually exclude such a thing. The question is, what causes human qualia? The actual experience of stimulus, rather than merely being a system reacting to them in a certain way.

The only way I can understand someone rejecting this assumption is if they somehow do not actually experience qualia.

I don’t think that’s the case at all. No assumption is made as to the nature of the difference. All sorts of explanations are offered, from “emergent behavior” to supernatural phenomena.

There is no such "hidden assumption" - the question is just literally asking what the difference is between the non-conscious program and the conscious one (in your case).